Rivets may be used for various fastening applications, such as attaching aircraft skin to aircraft frame components, various interior structures of aircraft, and the like. Rivets are widely used and tend to be one of the lightest and least expensive methods of fastening structural components together. However, inserting rivets and buckling/upsetting their shank ends (to form rivet tails) can be a very labor intensive process and can require highly skilled operators. One processing aspect that can be particularly challenging is aligning two dies of a riveting tool relative to the rivet. During riveting operation, one die supports the rivet head, while the other die forms the rivet tail. Both dies and the rivet need to be strictly coaxial to ensure proper formation of the rivet tail. This alignment difficulty drives up the cost of riveting joints and tend to offset the inherent low cost of riveting materials. Quality of riveted joints depends on the die-rivet alignment. Self-aligning riveting tools and methods of installing rivets using such self-aligning riveting tools are needed to reduce labor intensity and ensure consistent die-rivet alignments.